4.2 Numerical Studies
4.2.1 Parametric Variations
In this section, the results of various parametric studies that were performed with the numerical model are presented. The following parametric study results are partially published in Oberdorfer et al.[2013b]. Here, the Double-U model is used because of its worldwide relevance in applications and especially in Europe. The main question of the studies is about how the efficiency of a BHE can be improved. In general, the criterion for the efficiency is the borehole’s thermal resistance Rb. This value accumulates all thermal resistances between the working fluid and the subsurface and, thus, a decreased Rb results in an improved thermal contact.
Generally, the experimental in-situ determination ofRbis done after evaluating the effec-tive thermal conductivityλef f from Equation2.23and replacingTb using Equation2.24.
Due to this derivation,Rb is only accurate within the line source approximation limita-tions, as discussed in Section 2.3. One advantage of the numerical model is that there is full access to all state variables and, thus, Rb can be determined exactly due to its definition as the temperature difference between working fluid and subsurface, divided
by the heat flow rate between the surfaces of the pipes and the BHE. The temperatures are calculated numerically by integration over the whole borehole surface and integra-tion over the pipes volumes, respectively. The numerical approach offers the possibility of performing changes of any desired parameter and of studying the effect on the BHE performance in terms ofRb.
A benchmark model run was performed using the parameters shown in Table 4.1, that correspond to the values from the experimental test site described in Section 3.1. The model run conditions are analogous to a thermal response test: Heat is injected at a constant rate of 4.9 [kW] for 24 [h]. The heat injection rate is controlled by measuring the mean BHE fluid outflow temperature and adapting the corresponding inflow temperature in every numerical time step.
The results are presented in terms of the effects of parameter changes on the borehole’s thermal resistance. In every figure, the change of Rb due to a parameter variation is shown. Rb and the corresponding parameter are divided by the benchmark model values to make the different studies comparable (relative quantities are marked with asterisks: R∗b =Rb/Rb,benchmark). The evaluated value ofRb from the benchmark run is Rb = 0.09 [mKW−1]. This is in the range of the TRT evaluation results from the test site, where the values for the three BHEs vary between 0.071-0.117 [mKW−1] (see also Table3.2).
Parameter Reference value Min Max Unit
Borehole depth L 70 7 140 [m]
Pipe radius rp 13.1 7.86 26.2 [mm]
Pipe distance d 61 30.5 103.7 [mm]
Borehole radius rb 95 76 228 [mm]
Fluid therm. cond. λf 1.71 0.1 10 [W m−1K−1]
Pipe therm. cond. λp 0.28 0.1 10 [W m−1K−1]
Grout therm. cond. λg 2.00 0.1 10 [W m−1K−1]
Table 4.1: Parameters of the benchmark model run and variation ranges of the para-metric studies
The thermal conductivities of the working fluid, the pipe material and the grout filling are varied and the impact on R∗b is shown in Figure 4.16. All of the resulting curves indicate a reciprocal relationship between the particular conductivity and the resistance which is consistent, because every individual element that is toggled between the working
0 1 2 3 4 5 6 0.5
1 1.5 2 2.5
λ*
R
b*
Figure 4.16: Thermal resistance in dependence of the involved domain thermal con-ductivities; Working fluid (red), pipe material (blue) and grout filling (black)
fluid and the subsurface can be seen as part of a serial resistance network2, and of course, the thermal resistance goes towards infinity if any one of the involved conductors of the network series goes towards zero.
The highest impact is observed at the thermal conductivity of the grout material. The joint resistance is significantly reduced when this parameter is increased above the ref-erence value and causes a heavy increase ofR∗b when it is decreased below the reference.
This strong effect ofλ∗g is assumed to be due to its high amount of thermal mass com-pared to other elements in a BHE. The thermal conductivity of the working fluid λ∗f has only a very low impact on R∗b. The reason for this is the fact that the major part of heat transport in the pipes is advective because of the turbulences of the pipe flow and, thus, the diffusive part is rather negligible. As is to be expected, the influence of the pipe material thermal conductivity λ∗grout lies between the other two.
The parametric study results are fitted to analytical functions to survey the inverse relationship between the thermal conductivities and R∗b. The results are outlined in Table 4.2. The fit results underline the plausibility of the assumption of a reciprocal linear relationship between the single thermal conductors and the joint resistance of the BHE. This can be concluded because the exponentsβ3from the polynomial fits are close
2As it is described and used e.g. byBauer[2011]
d d
rb rpipe
L
Figure 4.17: Geometric quantities of the BHE that are varied in the study
to 1 and the Mean Squared Error (MSE) value is low. However, β3 is slightly smaller in all three cases and, thus, there must be processes that overlap the pure reciprocal relationship. An explanation can be found in the fact that the down-flow and up-flow pipes thermally interact with each other and cause thermal short cuts.
Figure4.18 shows results from parametric studies where the dependence ofR∗b on three selected geometric quantities is determined; the BHE length L, the borehole radius rb and the distance between the pipes d are varied. The black data points represent the dependence of the thermal resistance on the length of the BHE. The values of R∗b do not change whenLis varied. The reason for this is that the heat injection rate per unit lengthq is kept constant in all cases. This is an interesting result because even though the radial heat flux through the BHE is not constant in axial direction due to different borehole temperatures in different depths, as shown in Section 4.1.1, the mean value of
Thermal conductivity β1 β2 β3 MSE
λ∗f 0.94 0.06 0.89 1.9e-10
λ∗p 0.63 0.37 0.96 1.1e-06
λ∗g 0.38 0.62 0.94 3.3e-06
Table 4.2: Fit results of the parametric studies in Figure4.16to the reciprocal linear functionR∗b =β1+β2λ−β3.
the thermal borehole resistance per unit length is independent of the borehole length.
This means that the axial heat flux profile is stretched when the borehole is deeper, but it does not change its shape.
The influence of the borehole radius on the borehole’s thermal resistance is represented by the blue data points. rb has a lower limit, the borehole radius must at least be large enough so that the pipes are still entirely inside the BHE. Theoretically, there is no upper limit for the borehole radius, but of course there are practical restrictions.
Rb increases with increasing rb because of the additional thermal-resistive grout mass between the pipes and the subsurface, but the curve slope decreases for larger borehole radii.
The distance between the BHE pipes d is varied within the possible range inside the borehole. d must at least be large enough to avoid touching of the single pipes and, furthermore, the proper construction of a numerical mesh between the pipes must be granted. The upper limit is the value for d right before the pipes reach the borehole boundary. The red curve in Figure 4.18 shows the effect of the distance of the pipes inside the borehole. An increased distance decreases Rb considerably, it is only half as large when the two extreme cases, pipes in contact with each other and pipes outside close to the borehole surface, are compared. This result matches to the conclusions from the work of Acu˜na and Palm [2009] who solved the BHE heat transfer problem numerically for two dimensional BHE cross sections and found that “the best U-pipe BHE configuration corresponds to when the pipes are completely apart from each other”.
Although Single-U pipes were examined for their study, the results are comparable due to the symmetry of these two problems.
In general, the heat pipes for closed loop geothermal applications are only available at a very limited number of different diameters which is due to the limited number of industrial production standards in this area. Therefore, there only exist experimental results and data for the few available sizes of pipes. The numerical investigation offers the possibility of having a free choice concerning the pipe radii. One only has to consider the following restrictions: The pipes must be thick enough to provide a reasonable meshing inside and, furthermore, it makes sense to make the pipes not too thin to keep the pressure drop between the in- and outlets at a reasonable value. Therefore, the pipe
0 0.5 1 1.5 2 2.5 0.8
1 1.2 1.4 1.6 1.8 2
L*, r
b
*, d*
R
b*
BHE Length L*
Borehole Radius r
b* BHE Pipe Distance d*
Figure 4.18: Thermal resistance in dependence of different geometric BHE propor-tions
radius is varied between 0.6−2·rp for this study and the impact onR∗b is shown in Figure 4.19.
The curve progression of the resulting reliance between rp∗ and R∗b is very interesting:
For small radii, the thermal resistance decreases with increasing radius due to (i) the increasing surface of the pipes which causes more heat exchange area and (ii) the smaller distance between the pipes to the borehole boundary. The latter is just a comparable effect to an increasing pipe distance, as discussed in the study above. However, if rp∗ is further increased, R∗b also increases again slightly, resulting in a minimum of R∗b at rp∗ ≈1.7·rp ≈22.3[mm]. A closer look at the heat flux circumstances helps to explain this local minimum of R∗b. The integrated normal heat fluxes through the pipe walls and the borehole are plotted in Figure4.20. As already mentioned, the radial pipe heat flow rises with increasing pipe radius. Yet, the radial heat flow through the borehole surface does not change significantly. The heat flow through the pipes becomes larger than the heat flow between BHE and subsurface, and this is only possible if heat flows directly between the up- and down-flow branches of the pipes. This arising thermal short cut is fatal for the system performance and leads after all to a rising borehole thermal resistance.
0.5 1 1.5 2 0.8
0.9 1 1.1 1.2 1.3
r
pipe*
R
b*
Figure 4.19: Thermal resistance in dependence of the pipe radius
0.5 1 1.5 2
2000 2500 3000 3500 4000 4500
r
pipe*
Q [W]
BHE Wall pipe Wall
Figure 4.20: Integral heat flux trough the pipe walls and through the borehole wall in dependence of the pipe radius